The velocity of a particle of mass 1 kg is given by `v=10sqrtx` the work done by the forceacting on the particle during its motion from `x=4` to `x=9m` is

Options

A. `250`J
B. `300`J
C. `350`J
D. `400`J

Answer

To solve the problem, we need to determine the work done by the force acting on a particle as it moves from \( x = 4 \) m to \( x = 9 \) m. The velocity of the particle is given by the equation \( v = 10\sqrt{x} \).

### Step-by-Step Solution:

1. **Identify the given parameters**:
- Mass of the particle, \( m = 1 \) kg
- Velocity, \( v = 10\sqrt{x} \)

2. **Calculate acceleration**:
- We know that acceleration \( a \) can be expressed as:
\[ a = v \frac{dv}{dx} \]
- First, we need to differentiate the velocity with respect to \( x \):
\[ v = 10\sqrt{x} \implies \frac{dv}{dx} = 10 \cdot \frac{1}{2\sqrt{x}} = \frac{5}{\sqrt{x}} \]
- Now, substituting \( v \) and \( \frac{dv}{dx} \) into the acceleration formula:
\[ a = 10\sqrt{x} \cdot \frac{5}{\sqrt{x}} = 50 \text{ m/s}^2 \]

3. **Calculate the force**:
- Using Newton's second law, \( F = ma \):
\[ F = 1 \text{ kg} \cdot 50 \text{ m/s}^2 = 50 \text{ N} \]

4. **Calculate the displacement**:
- The displacement \( d \) when moving from \( x = 4 \) m to \( x = 9 \) m is:
\[ d = x_f - x_i = 9 \text{ m} - 4 \text{ m} = 5 \text{ m} \]

5. **Calculate the work done**:
- Work done \( W \) is given by the formula:
\[ W = F \cdot d \]
- Substituting the values we found:
\[ W = 50 \text{ N} \cdot 5 \text{ m} = 250 \text{ J} \]

### Final Answer:
The work done by the force acting on the particle during its motion from \( x = 4 \) m to \( x = 9 \) m is \( 250 \) Joules.

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